Question: Simplify the following expression: $ k = \dfrac{x - 3}{x - 2} - \dfrac{9}{2} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{2}{2}$ $ \dfrac{x - 3}{x - 2} \times \dfrac{2}{2} = \dfrac{2x - 6}{2x - 4} $ Multiply the second expression by $\dfrac{x - 2}{x - 2}$ $ \dfrac{9}{2} \times \dfrac{x - 2}{x - 2} = \dfrac{9x - 18}{2x - 4} $ Therefore $ k = \dfrac{2x - 6}{2x - 4} - \dfrac{9x - 18}{2x - 4} $ Now the expressions have the same denominator we can simply subtract the numerators: $k = \dfrac{2x - 6 - (9x - 18) }{2x - 4} $ Distribute the negative sign: $k = \dfrac{2x - 6 - 9x + 18}{2x - 4}$ $k = \dfrac{-7x + 12}{2x - 4}$